Abstract
We use the Suita conjecture (now a theorem) to prove that for any domain \(\Omega \subset \mathbb {C}\) its Bergman kernel \(K(\cdot , \cdot )\) satisfies \(K(z_0, z_0) = \hbox {Volume}(\Omega )^{-1}\) for some \(z_0 \in \Omega \) if and only if \(\Omega \) is either a disk minus a (possibly empty) closed polar set or \(\mathbb {C}\) minus a (possibly empty) closed polar set. When \(\Omega \) is bounded with \(C^{\infty }\)-boundary, we provide a simple proof of this using the zero set of the Szegö kernel. Finally, we show that this theorem fails to hold in \(\mathbb {C}^n\) for \(n > 1\) by constructing a bounded complete Reinhardt domain (with algebraic boundary) which is strongly convex and not biholomorphic to the unit ball \(\mathbb {B}^n \subset \mathbb {C}^n\).
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The authors sincerely thank László Lempert, Song-Ying Li, and the anonymous referee for the valuable comments on this paper.
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Dong, R.X., Treuer, J. Rigidity Theorem by the Minimal Point of the Bergman Kernel. J Geom Anal 31, 4856–4864 (2021). https://doi.org/10.1007/s12220-020-00459-2
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DOI: https://doi.org/10.1007/s12220-020-00459-2